Queueing Systems 8 (1991) 211-224 211 REGENERATION AND RENOVATION IN QUEUES S.G. FOSS Novosibirsk State University, Pirogova 2, 630090 Novosibirsk, USSR V.V. KALASHNIKOV Institute for Systems Studies, 9, Prospect 60 let Oktjabrja, 117312 Moscow, USSR Received 17 May 1990; revised 24 November 1990 Regenerative events for different queueing models are considered. The aim of this paper is to construct these events for continuous-time processes if they are given for the corresponding discrete-time model. The construction uses so-called renovative events revealing the property of the state at time n of the discrete-time model to be independent (in an algebraic sense) of the states referring to epochs not later than n- L (where L is some constant) given that there are some restrictions on the "governing sequence". Different types of multi-server and multi-phase queues are considered. Keywords: Regeneration, renovation, stopping time, queueing system, multi-server system, multi-phase system. 1. Introduction The notion of regeneration is a very important one in probability theory in general and in queueing theory in particular. It is used both for qualitative analysis of queueing processes (their ergodicity, boundedness, stability, conver- gence to a stationary regime) and quantitative estimates including simulation (stationary characteristics, estimates of stability, convergence rates etc.). Being introduced by W.L. Smith this notion was generalized by Thorisson [13] and Asmussen [1] in such a way that they permitted regeneration cycles to be dependent given that any cycle is independent of the preceding regeneration times. This generalization is very important for queueing theory as it preserves all important results and leads to the possibility of considering a rather wide class of queues (in comparison with Smith's regeneration). This generalization is very natural. In fact, different examples of it were considered independently by other authors; see e.g. Kalashnikov [9]. Another (but not totally different) approach was suggested by Borovkov ([3] for so-called stochastic recursive sequences. Its essence is the construction of so-called "renovative events" with the following property: the process is indepen- J.C. Baltzer A.G. Scientific Publishing Company 212 S. G. Foss, V. V. Kalashnikov / Regeneration and renovation dent (in an algebraic sense but not necessarily a probabilistic one!) of the preceding terms of the governing sequence after the time when some renovative event does occur. It turned out that "coupling" of these two approaches permits us to construct regenerative events in some new situations (e.g. for multi-server and multi-phase queues). The aim of this paper is to suggest a general construction of regeneration points for continuous-time models given that they exist for some imbedded discrete-time process. Now the construction of regeneration points for discrete- time processes is well-known (see Thorisson [13], Asmussen [1], Kalashnikov [9] and references in these papers). As a rule it exploits the Harris-recurrency of the underlying Markov processes, see e.g. Nummelin [12]. But its generalization on continuous time is not self-evident. Here we propose it for rather general processes occurring in queueing. So, it can be used for many queueing models. We do it step by step, starting from the discrete case, using such notions as regeneration and renovation. Given that regeneration points are constructed it is possible to apply any known result for regenerative processes (conve.rgence rate estimates, stability estimates, estimates of distribution functions of the first-occurrence times, etc.) in order to study the corresponding queueing model. An example of such a study is contained in Asmussen and Foss [2] where the proposed general construction is used for obtaining ergodicity results. The idea of this paper arose after fruitful discussions with S. Asmussen and H. Thorisson during the meeting on queueing theory and point processes in Karpacz, Poland (January, 1990). 2. Regeneration and renovation in discrete time Regeneration events in queueing are often connected with emptiness of queues, or arrival of customers to an empty system. Example 1: The single-server queue Let us consider the GI/GI/1/~ queue which satisfies Lindley's equation: w,+l=(w,+s,-e,)+, n>~O, (2.1) where (-) + = max(0, ), and { x, }, { e, } are sequences of service and interarrival times respectively consisting of i.i.d.r.v.'s, and { w,,} is a sequence of actual waiting times. Here and below, interarrival and service times are numbered from 0: the zeroth customer having service time s o arrives at time e 0 and so on. Then the event A.= (w.=O} (2.2) S. G. Foss, V. V. Kalashnikov /Regeneration and renovation 213 is a regenerative one. Let us denote S(k) the kth occurrence time of the regenerative event, Ok = S(k) - S(k - 1), k >/1, S(0) = 0. If Es o < Ee o (2.3) then the events ( A, } are positive recurrent. This means that under the condition (2.3) the following relation is true EOk <~ e < oo, k >~ l, (2.4) where the constant c depends on the distribution functions (d.f.) of s o and e 0 in general. Let us notice that the inequality (2.3) follows from P(s o < e0) > 0. (2.5) Sometimes (e.g. in stability analysis, see Borovkov [4], Kalashnikov [8], Kalashnikov and Rachev [11]) we need this constant to be the same for some class of regenerative processes. Then we have to demand the restrictions to be uniform over this class (in some natural sense). This example shows that the process ( wn } is a regenerative one in the sense of Smith, i.e. the "fragments" of this process belonging to different cycles are independent. The "condition" of regeneration is expressed here in pure algebraic form (2.2). Example 2: Multi-server queues Consider a multi-server system (with N servers) and use for its description Kiefer-Wolfowitz equations: w,+l=R(w,+~s,-Ie,) +, n>~O, (2.6) where again (s,} and (e,} are sequences of service and interarrival times, w, = (w,1 .... , W,N ) is a waiting time vector referring to the nth customer, Wnl ... ~< w,~, and R(-) is an operator which orders the components of (-) in a non-decreasing way, 8 = (1, 0,..., 0), I = (1, 1,..., 1). If ( s, } and { e, } consist of i.i.d.r.v.'s then it is possible to define regenerative events for this system. Very often one tries to expand the previous construction to the multi-server case in the following way. Let A, = { w, = (0, 0 ..... 0) }. (2.7) Of course, this is a regenerative event in the sense of Smith. In order for these events to be positive recurrent we need to impose the ergodicity condition: Es o < NEe o. (2.8) But this condition is not sufficient in general. We have to demand additionally that P(s o < e0) > 0. (2.9) It is easy to prove that under conditions (2.8) and (2.9) the events {A,} constructed by formula (2.7) are positive recurrent. 214 S. G. Foss, V. V. Kalashnikov / Regeneration and renovation This construction is not good because it demands an additional restriction (2.9). A similar restriction (2.5) for a single-server queue was implied by ergodic- ity condition (2.3) and so it was not an additional one. It is well-known (see e.g. Borovkov [3], Kalashnikov [9], Foss [6], Kalashnikov and Rachev [11]) that it is possible to eliminate the restriction (2.9) if we use regeneration in the sense of S. Asmussen and H. Thorisson. Namely, let us fix an integer L > 0 and consider the events B.(A, o, e) = {sj-Nej< --A, sj<o, ej>~e, n-L<j<n}, (2.10) C,(W) = {w,u~< W}, (2.11) where A, e, e, W are some positive constants. It is possible to prove that there exists such integer L > 0 that the values wn (n >/L) calculated with the help of eq. (2.6) do not depend on w0 .... ,wn_ L, (but only on e,_L .... ,e,_l and s,-L .... , G-l) in an algebraic sense given that the event A,= G_L(W)nB,(A, O, e) (2.12) does occur. In fact, this is a consequence of the Kiefer-Wolfowitz equation. Of course, the constant L depends on W, A, o, e and it is possible to give a corresponding estimate, see Kalashnikov and Rachev [11]. It is reasonable to name A, a "renovative event". Let us write I(A) = 1 if an event A does occur and I(A) = 0 otherwise. If we denote S(0) = 0, S(1) = rain(k: I(Ak) = 1}, (2.13) S(n+ 1)= min(k: k> S(n)+L, I(AI,) = 1}, (2.14) then the sequence (S(n)},>_.a of "renovation epochs" is a renewal one, i.e.r.v.'s On = S(n) - S(n - 1) are i.i.d. (n > 1). Besides, the ergodicity condition (2.8) implies that EO 1 < o~. So, we managed to eliminate condition (2.9) and obtained a regenerative process without it. Though inter-regeneration times are i.i.d.r.v.'s, the cycles are dependent in general. Namely, the behaviour of the process (w, } beginning from some epoch S(k) can depend on the values Ss(k)_L,..., SS(k)_ 1 and es(k)_ L .... ,es(~)_ a and, hence, on the values WS(k)_L+X,...,Ws(k)_ 1. Of course, the distribution of the process (w~k)},>~0- (W,+s(k)},>_. 0 does not de- pend on (S(O),...,S(k)} for any k. This means that the process (w,} is a regenerative one in the sense of S.